Geometric properties and asymptotic behavior of solutions of diffusion equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 23340035
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tohoku University
• Principal Investigator: Ishige Kazuhiro 東北大学, 理学(系)研究科(研究院), 教授 (90272020)
• Co-Investigator(Kenkyū-buntansha): KOZONO Hideo 早稲田大学, 理工学術院, 教授 (00195728) ; OGAWA Takayoshi 東北大学, 大学院理学研究科, 教授 (20224107) ; KOIKE Shigeaki 東北大学, 大学院理学研究科, 教授 (90205295) ; YAMADA Sumio 学習院大学, 理学部, 教授 (90396416)
• Co-Investigator(Renkei-kenkyūsha): YANAGIDA Eiji 東京工業大学, 理工学研究科, 教授 (80174548) ; KABEYA Yoshitsugu 大阪府立大学, 大学院工学研究科, 教授 (70252757)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: 放物型方程式 / 形状解析 / 漸近解析 / 爆発集合 / 放物型凸 / 最大点挙動 / 動的境界条件

Outline of Final Research Achievements

We developed the method for studying geometric properties and asymptotic behavior of solutions of parabolic equations, and obtained the asymptotic behavior of hot spots and the optimal decay rates of the Lebesgue norms for the heat equation with a potential. Furthermore, we established a method of obtaining the higher order asymptotic expansions of the solutions behaving like the heat kernel. In addition, we study the location of the blow-up set for a semilinear heat equation by the profile of the solution just before the blow-up time. In particular, we gave a sufficient condition for no boundary blow-up.

Free Research Field

偏微分方程式論